To summarize, the mathematical definitions of dimensions do not embody any physical notions. Size and powers are quantitative parameters that are studied exclusively within Mathematics. In Physics, there are only intuitive relations of size – big and small – and of orientation – orthogonal, left/right, parallel, or three- dimensional. When we say that a house is 3 times larger than an elephant, we are using the elephant as a measuring stick and counting the number of times the elephant fits into the house. This is not Physics. Old Father Universe never learned to count and doesn't understand 'three'. He only understands elephant and house. Physics doesn't deal with quantitative relations involving observers and an artificial language called Math because if an ET has a different measuring stick, he reaches a different conclusion about our Universe. The difference between Physics and Math is that Physics is universal and intuitive. In Physics there are no ludicrous and artificial notions such as spin 0, ½, 1, and 2. In Physics there is only intuitive clockwise and counterclockwise. In Physics there is no such nonsense called energy. There are only things, each one of which has shape. And in Physics we do not gauge the size of the width of a box with a number line. In Physics length, width, and height have the same 'size,' and all things, including the box, are three-dimensional. It never goes beyond this. It can't. Physics is only qualitative. The mathematical definitions of dimension would be of no consequence if they were used contextually. As shown in the foregoing examples, mathematicians have blended these irreconcilable definitions into a collage they incongruously call Mathematical Physics.
In the religion of relativity, we use one finger to designate direction and four when we refer to dimension. Got it, Bill?